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Chapter 0: Problem 125

Describe ways in which solving a linear inequality is similar to solving a linear equation.

Short Answer

Expert verified
Solving linear inequalities and equations are similar in that they both involve variables, constants, arithmetic operations, can be graphically represented with a line, and solved using addition, subtraction, multiplication, and division. The difference lies in the interpretation of the solution- a certain point for equations and a range of values or area for inequalities.

Step by step solution

01

Identify Basic Properties

Linear equations and linear inequalities share several basic properties. They both involve variables, constants, and arithmetic operations. Both can be represented graphically using a line on the coordinate plane.
02

Understand Similar Solving Methods

Linear equations and inequalities can both be solved using similar strategies including addition, subtraction, multiplication, and division of the same or different numbers from both sides. Also both the expressions can be simplified by combining similar terms on the same side.
03

Recognize Differing Symbol Interpretations

While both enjoy similar processes of algebra manipulation, the way solutions are interpreted differs. In equations, we use \(=\) (equal to), meaning the solution is a certain specific value or point on the graph. In inequalities, we use symbols such as \(\>\), \(\<\), \(\leq\), or \(\geq\), meaning the solution is a range of values or area on the graph.

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Most popular questions from this chapter

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{2}+\frac{2}{3}$$

Explain how to factor \(3 x^{2}+10 x+8\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)

Will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\)

Exercises \(159-161\) will help you prepare for the material covered in the next section. In parts (a) and (b), complete each statement. a. \(\frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{2}\) b. \(\frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?}\) c. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?
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